An acyclic edge coloring of a graph is a proper edge coloring in which there are no bichromatic cycles. The acyclic chromatic index of a graph $G$ denoted by $a'(G)$, is the minimum positive integer $k$ such that $G$ has an acyclic edge coloring with $k$ colors. It has been conjectured by Fiam\v{c}\'{\i}k that $a'(G) \le \Delta+2$ for any graph $G$ with maximum degree $\Delta$. Linear arboricity of a graph $G$, denoted by $la(G)$, is the minimum number of linear forests into which the edges of $G$ can be partitioned. A graph is said to be chordless if no cycle in the graph contains a chord. Every $2$-connected chordless graph is a minimally $2$-connected graph. It was shown by Basavaraju and Chandran that if $G$ is $2$-degenerate, then $a'(G) \le \Delta+1$. Since chordless graphs are also $2$-degenerate, we have $a'(G) \le \Delta+1$ for any chordless graph $G$. Machado, de Figueiredo and Trotignon proved that the chromatic index of a chordless graph is $\Delta$ when $\Delta \ge 3$. They also obtained a polynomial time algorithm to color a chordless graph optimally. We improve this result by proving that the acyclic chromatic index of a chordless graph is $\Delta$, except when $\Delta=2$ and the graph has a cycle, in which case it is $\Delta+1$. We also provide the sketch of a polynomial time algorithm for an optimal acyclic edge coloring of a chordless graph. As a byproduct, we also prove that $la(G) = \lceil \frac{\Delta }{2} \rceil$, unless $G$ has a cycle with $\Delta=2$, in which case $la(G) = \lceil \frac{\Delta+1}{2} \rceil = 2$. To obtain the result on acyclic chromatic index, we prove a structural result on chordless graphs which is a refinement of the structure given by Machado, de Figueiredo and Trotignon for this class of graphs. This might be of independent interest.

翻译：图$G$的无圈边着色是指一种正常边着色，使得图中不存在二色圈。图$G$的无圈边色指数记为$a'(G)$，是使得$G$存在使用$k$种颜色的无圈边着色的最小正整数$k$。Fiam\v{c}\'{\i}k猜想：对任意最大度为$\Delta$的图$G$，有$a'(G) \le \Delta+2$。图$G$的线性荫度记为$la(G)$，是将$G$的边划分成线性森林所需的最少森林数。若图中任何圈都不含弦，则称该图为无弦图。每个$2$-连通无弦图都是极小$2$-连通图。Basavaraju和Chandran证明：若$G$是$2$-退化图，则$a'(G) \le \Delta+1$。由于无弦图也是$2$-退化图，故对任意无弦图$G$有$a'(G) \le \Delta+1$。Machado、de Figueiredo和Trotignon证明：当$\Delta \ge 3$时，无弦图的边色指数为$\Delta$。他们还给出了最优着色无弦图的多项式时间算法。我们改进了这一结果，证明无弦图的无圈边色指数为$\Delta$，除非当$\Delta=2$且图包含圈时，此时为$\Delta+1$。我们还给出了无弦图最优无圈边着色的多项式时间算法框架。作为副产品，我们还证明$la(G) = \lceil \frac{\Delta }{2} \rceil$，除非$G$包含$\Delta=2$的圈，此时$la(G) = \lceil \frac{\Delta+1}{2} \rceil = 2$。为得到无圈边色指数的结果，我们证明了关于无弦图的结构性结论，该结论是对Machado、de Figueiredo和Trotignon针对此类图所给结构的精化，这可能具有独立的研究价值。